The conjugate symmetric property of an inner product states that $\langle{x, y}\rangle = \overline{\langle{y, x}\rangle}$. My question is regarding showing this when we don't necessarily know that our field is $\mathbb{C}$.
For a vector space $V$ defined over a field $\mathbb{F}$, let $\langle{,}\rangle: V \times V \rightarrow \mathbb{F}$ be the inner product defined by $\langle{u, v}\rangle = u^t\cdot{}v$.
Then $\langle{u, v}\rangle = u^t\cdot{v} = v^t\cdot{}u = \langle{v, u}\rangle$.
If the field was $\mathbb{R}$, this would be fine. But I'm not sure about the case that the field is $\mathbb{C}$.
I'm not sure how to just throw the conjugate in there. I'm not sure how to conclude that $\langle{u, v}\rangle = \overline{\langle{v, u}\rangle}$.